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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: 𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvdmsscn.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
Ref | Expression |
---|---|
dvdmsscn | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 17359 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 | |
2 | dvdmsscn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | 1, 2 | sselid 3976 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
4 | elpwi 4603 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | dvdmsscn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
7 | recnprss 25350 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 5, 8 | sstrd 3988 | 1 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3944 𝒫 cpw 4596 {cpr 4624 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 ℝcr 11091 ↾t crest 17348 TopOpenctopn 17349 ℂfldccnfld 20878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 ax-resscn 11149 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-rest 17350 |
This theorem is referenced by: dvxpaek 44429 etransclem17 44740 etransclem18 44741 etransclem20 44743 etransclem21 44744 etransclem22 44745 etransclem29 44752 etransclem31 44754 etransclem34 44757 etransclem43 44766 etransclem46 44769 |
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